Each person starts working life on a salary of 5000 dollars and then benefits form an annual increment of 250 dollars over 40 years of his career.

Question

I have this very simple math question: Each person starts working life on a salary of $5000$ dollars and then benefits form an annual increment of $250$ dollars over $40$ years of his career.

My work: So in

1. First year he gets $5000$ dollars,
2. Second year gets $5000+(250)$ dollars
3. Third year gets $5000+2(250)$ dollars, and so on till $40$ years

How can I get his total income after $40$ years? Is it just $40*250+5000$ dollars?

Answer 1

Assuming that he starts his working life in year $0$ and then works through years $1,2, \ldots, 40$ his total income is equal to $$\sum_{n=0}^{40}\left(5000+n\cdot250 \right)=41\cdot5000+250\sum_{n=1}^{40}n=41\cdot5000+250\frac{40\cdot(40+1)}{2}=410000$$ where the last equation was obtained by the formula for summation of consecutive integers.